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Author: William A. Stein
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\documentclass{article}
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\usepackage{fullpage}
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\pagestyle{empty}
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\usepackage{amsmath}
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\renewcommand{\cong}{\equiv}
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\title{$U_p\left(\frac{1}{j}\right)$}
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\author{William Stein}
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\date{December 2003}
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%\voffset=-0.05\textheight
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%\textheight=1.1\textheight
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\hoffset=-0.03\textwidth
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%\textwidth=1.5\textwidth
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\begin{document}
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\maketitle
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\small
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\begin{align*}
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U_{5}\left(\frac{1}{j}\right) & \cong{} 0 \pmod{5}\\
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U_{5}\left(\frac{1}{j}\right) & \cong{} 940\left(\frac{1}{j}\right) \pmod{5^5}\\
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U_{5}\left(\frac{1}{j}\right) & \cong{} 10315\left(\frac{1}{j}\right) + 3125\left(\frac{1}{j}\right)^2 \pmod{5^6}\\
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U_{5}\left(\frac{1}{j}\right) & \cong{} 744690\left(\frac{1}{j}\right) + 1596875\left(\frac{1}{j}\right)^2 + 3781250\left(\frac{1}{j}\right)^3 \pmod{5^{10}}\\
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U_{5}\left(\frac{1}{j}\right) & \cong{} 20275940\left(\frac{1}{j}\right) + 21128125\left(\frac{1}{j}\right)^2 + 13546875\left(\frac{1}{j}\right)^3 + 9765625\left(\frac{1}{j}\right)^4 \pmod{5^{11}}\\
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U_{5}\left(\frac{1}{j}\right) & \cong{} 18819104065\left(\frac{1}{j}\right) + 1730112500\left(\frac{1}{j}\right)^2 + 29066281250\left(\frac{1}{j}\right)^3 + 1523437500\left(\frac{1}{j}\right)^4 + 20800781250\left(\frac{1}{j}\right)^5 \pmod{5^{15}}\\
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U_{5}\left(\frac{1}{j}\right) & \cong{} 49336682190\left(\frac{1}{j}\right) + 428976206250\left(\frac{1}{j}\right)^2 + 273206906250\left(\frac{1}{j}\right)^3 + 306699218750\left(\frac{1}{j}\right)^4 + \\
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& \qquad 539599609375\left(\frac{1}{j}\right)^5 + 640869140625\left(\frac{1}{j}\right)^6 \pmod{5^{17}}\\
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U_{5}\left(\frac{1}{j}\right) & \cong{} 49336682190\left(\frac{1}{j}\right) + 26368917612500\left(\frac{1}{j}\right)^2 + 90300062375000\left(\frac{1}{j}\right)^3 + 42268369140625\left(\frac{1}{j}\right)^4 + \\
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& \qquad 33345996093750\left(\frac{1}{j}\right)^5 +
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83801269531250\left(\frac{1}{j}\right)^6 + 77056884765625\left(\frac{1}{j}\right)^7 + \pmod{5^{20}}\\
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U_{5}\left(\frac{1}{j}\right) & \cong{} 49336682190\left(\frac{1}{j}\right) + 1266145528940625\left(\frac{1}{j}\right)^2 + 1711546400265625\left(\frac{1}{j}\right)^3 + 900575253906250\left(\frac{1}{j}\right)^4 + \\
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& \qquad 1940694628906250\left(\frac{1}{j}\right)^5 +
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2372619628906250\left(\frac{1}{j}\right)^6 + 363159179687500\left(\frac{1}{j}\right)^7 + 2002716064453125\left(\frac{1}{j}\right)^8 \pmod{5^{22}}\\
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%U_{5}\left(\frac{1}{j}\right) & \cong{} 49336682190\left(\frac{1}{j}\right) + 118091249288706250\left(\frac{1}{j}\right)^2 + 85158049085812500\left(\frac{1}{j}\right)^3 + 12821504208984375\left(\frac{1}{j}\right)^4 + 216517415820312500\left(\frac{1}{j}\right)^5 + 293243286132812500\left(\frac{1}{j}\right)^6 + 162487792968750000\left(\frac{1}{j}\right)^7 + 109291076660156250\left(\frac{1}{j}\right)^8 + 164508819580078125\left(\frac{1}{j}\right)^9 + O(\left(\frac{1}{j}\right)^11) \pmod{5^25}\\
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\end{align*}
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The $5$-adic valuations of the coefficients of the power series $U_5(1/j)$ are
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$$ 1, 5, 6, 10, 11, 15, 17, 20, 22, 25, 26, 30, 31, 35, 36, 40, 43, 46,\ldots. $$
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On the other hand, if $p=7$, then the $7$-adic valuations of the
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coefficients of $U_7(1/j)$, written in terms of $1/j$ are
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$$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ldots.$$
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Note that $0$ is not a supersingular $j$-invariant in characteristic~$7$.
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When $p=11$, the invariant $0$ is supersingular, but $1$ is
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also supersingular.
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The $11$-adic valuations of the coefficients of $U_{11}(1/j)$, written
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in terms of $1/j$, are
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$$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ldots.$$
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\end{document}